I was trying to work out the group of symmetries of a square pyramid and obtained a group of 8 elements of which the four rotations about the base square are orientation preserving.
All 4 of the non-orientation preserving isometries of the square and the square pyramid are basically obtained by reflections along a plane perpendicular to the axis of rotation. Such as, the planes which pass through the two diagonals of the base square and the lines joining midpoints of opposite sides of the square.
Though apparently these groups are distinct when I looked it up. ($C_{4v}$ for the square pyramid and $D_4$ for the square) I don't see what distinguishes the group of symmetries of the square pyramid from that of the square... Am I missing something here?
They are isomorphic groups, but distinct in that isometries of the plane are not the same thing as isometries of 3-space.
For instance, a reflection in $D_4$ could correspond to either a reflection or a rotation acting on a plane embedded in 3-space. (In general, any reflection in $n$-space can be realized by a rotation when the space is embedded in a higher-dimensional space.)